R-L ALGORITHM: AN APPROXIMATION ALGORITHM FOR FRACTAL SIGNALS BASED ON FRACTIONAL CALCULUS
CHENGYUAN WANG, Huobin Tan, Jiakui Hu, DULAN ZHOU · Fractals · 2021
An important characteristic of a fractal signal is that its graph is not smooth in any small interval. This indicates the difficulty of the approximation of fractal signals, because traditional approximation methods normally require some certain smoothness of the approximated function. However, recent studies have shown that fractal functions that satisfy the Hölder condition can be linearly changed in the fractal dimension of their graphs by the fractional calculus, which implies that we can use the fractional calculus to make graphs of fractal signals smoother, and then approximate these fractal signals. This paper first gives our research background and related theories of fractals and the fractional calculus, and then introduces the main research content, including the following aspects: (1) Proposing a fractal signal approximation algorithm, the R-L algorithm, explaining and deriving how to implement this algorithm. (2) Aiming at a specific fractal signal, carrying out an approximation experiment, and confirming the R-L algorithm is better than a direct approximation. (3) According to the experimental process and conclusions, introducing our future work, such as further optimization of the R-L algorithm, and showing the R-L algorithm may have the predictive capability outside the sampling interval. The R-L algorithm can effectively modify the smoothness of graphs of the fractal signal, so that more types of approximation algorithms can be selected with a better approximation effect.